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Existing score-based adversarial attacks mainly focus on crafting top-1 adversarial examples against classifiers with single-label classification. Their attack success rate and query efficiency are often less than satisfactory, particularly under small perturbation requirements; moreover, the vulnerability of classifiers with multilabel learning is yet to be studied. In this paper, we propose a comprehensive surrogate free score-based attack, named geometric score-based black-box attack (GSBAK), to craft adversarial examples in an aggressive top-K setting for both untargeted and targeted attacks, where the goal is to change the top-K predictions of the target classifier. We introduce novel gradient-based methods to find a good initial boundary point to attack. Our iterative method employs novel gradient estimation techniques, particularly effective in top-K setting, on the decision boundary to effectively exploit the geometry of the decision boundary. Additionally, GSBAK can be used to attack against classifiers with top-K multi-label learning. Extensive experimental results on ImageNet and PASCAL VOC datasets validate the effectiveness of GSBAK in crafting top-K adversarial examples.